A field theory formulation of polymer networks

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A field theory formulation of polymer networks

S.F. Edwards

To cite this version:

S.F. Edwards. A field theory formulation of polymer networks. Journal de Physique, 1988, 49 (10),

pp.1673-1682. �10.1051/jphys:0198800490100167300�. �jpa-00210848�

1673

A field theory formulation of polymer ^networks

S. F. Edwards

Cavendish Laboratory, Madingley Road, Cambridge ^{CB3 OHE, G.B.}

(Reçu le 11 décembre 1987, révisé le 20 mai 1988, accepté ^{le 16} juin 1988)

Résumé.

De nouveaux types de réseaux sont apparus récemment ^avec des fonctionnalités et des lois de

probabilités inhabituelles. Pour les étudier, ^nous proposons ^{un nouveau} formalisme de théorie des champs

dans lequel le ^problème du calcul de l’énergie ^libre élastique ^se formule de manière concise.

Abstract.

New kinds of networks have appeared ^{in recent} years which have ^non standard functionalities and

non standard end to end probabilities ^{of the} segments. To handle these a new field theoretic formalism is

proposed, and it is shown that the problem ^of calculating the elastic free energy ^can be formulated in a concise way using the formalism.

J. Phys. ^{France 49} (1988) ^{1673-1682 OCTOBRE 1988, 1}

Classification

Physics ^Abstracts

03.20

03.70

81.20

1. Introduction.

The simplest ^{and most} familiar form of a cross linked network is the result of the historical experiment ^of introducing ^cross linking agents ^{into a} polymer melt,

or concentrated solution. Chains are then perma-

nently ^bonded, ideally ^{in random} positions, ^but very often in some way which reflects the history ^{of the} bonding. Putting aside that complication, ^{there are}

still many restrictions in the standard « vulcani- zation ^» model : (i) ^{the chains are} normally ^{taken as} Gaussian, ^{whereas one now has} liquid crystal poly-

mers being ^{used in networks} and various inter- mediate or mixed conditions ; (ii) ^{the cross} linking ⁱⁿ

a vulcanization will always ^{have a four fold} functionality, whereas many other functionalities are

available ; (iii) polymers ^of specific length ^distri-

bution can be made with the cross linkage being arranged ^{to take} place ^{at their ends, or other} specified positions, ^{so that a} network with a given

distribution of lengths ^{between cross links can be}

constructed.

If one takes the classic paper of James and Guth

on molecular networks (James ^{and Guth} [1]), ^{it is} perfectly possible ^{to include} all these extensions in their formalism, ^but it will lead to an extremely complex formalism since it carries far more infor- mation than is necessary. The far simpler ^formalism

of Wall, ^much developed by ^{Wall and} Flory [2], ^is

not comprehensive in the way that the James and Guth theory is, since it makes the approximation

that the cross link points ^deform affinely. Clearly

this fails for a liquid crystal polymer ^{network which,}

since the polymers ^are inextensible, ^{must have its}

entire entropy residing in the freedom of the cross

links to move.

For the case of cross linking ^{at random} points ^a

concise formulation exists in the replica ^formalism (Edwards ^{and Deam} [3]). ^This exploits the fact that it is possible ^{to find a} concise notation for a Gaussian molecule by ^{means of} the Wiener integral. ^{If a} polymer ^{whose monomer} points ^are Ro, Ri, ^..., Rm

is considered continuous so that the labels

0, 1, ..., M ^are replaced by ^{the arc} length ^s,

0 , s , L where Mf

L, f being ^the (Kuhn) step length, ^{then the} probability ^of finding ^a configuration R (s ) ^is

provided ^{that one does not look at} distances of the order of (or ^less than) ^f.

(The ^Wiener picture ^{was de} facto introduced into

polymer ^science by Rouse ; ^{see for} example ^{Doi and}

Edwards [3] who showed that the long ^wave dynamics ^of polymers ^was independent ^{of their}

shortwave structure). Cross links can be added by putting ^{in 8 functions} i.e. if the i-th chain Ri (si )

meets the j-th ^chain Rj (Sj) ^at s’ ^and sJ respectively

is introduced into the probability distribution.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490100167300

In order to calculate the free energy, ^one needs the free energy ^{as a} function of the si, F ( [s ] ) ^which

then has to be averaged ^over the si using ^the probability ^{P of} finding the s :

The particular simplification ^{of the} replica ^{method is}

based on the fact that

so that, ^since

where the partition function Z is given by

A particularly simple ^case is when P is just ^the probability ^of finding ^chains touching ⁱⁿ equilibrium

before strain, ^{when an} instantaneous cross linkage

takes place. ^{When this} happens

where Z is the value of Z when s can take all values,

i.e. of a pure slip ^link polymer in which the link can

slide along ^the length,

Fo the free energy of sliding ^as against permanent

cross linkage. ^{We then have}

or if we define F (n ) :

Note that Z (0) is calculated before strain, ^{zn after. It}

is well known that for homogeneous material, ^the

free energy after ^a change ^{in the} shape ^{of the}

material by ^a change ^{in the three Cartesian axes} by À l’ À 2’ À 3 gives ^a complete description of the elastic behaviour. To obtain F (,k 1, A 2, A 3 ) ^we proceed ^this

way :

we regard each of the Zn as lying ^{in its own three}

dimensional space, and hence imagine ^an initial box

(0) ^and (1)... (n ) replicas

where (0) is the unstrained network side, say unity,

with n replicas, ^{strained to sides} À 1’ À 2’ A 3. ^The

cross link points ^{are the same for all members of}

the replica system, ^{so at this} point they ^{can be} averaged ^over (which ^{is of course the whole} point ^of

this manoeuvre). The constraint of cross linkage ^is

now

which is conveniently exponentiated ^{via a} fugacity

: ^:

For a well crossed link network one can ignore ^end conditions, and consider all the polymer ^{as one} long

chain i.e. L is now the total length ^of polymer present. ^{Then we} finally ^reach

One can even go ^one stage further and introduce a vector fll in 3 n + 3 dimensions so that

:R = (R(O), R(1), ..., R(n»,

where the vector 3t lies in the space of diagram ^1.

Thus the Wiener representation ^{allows a theorem to}

be stated : the free energy of ^a polymer ^network

with random cross linkage ^{is identical with the}

1675

problem ^{of a} random walk which meets itself N times in 3 n + 3 dimensions in a specified space.

Details of this method are given in the references above and in Edwards and Vilgis [4] ; ^we quote ^the basic results because the formula (1.17) ^represents

the whole of the problem ^{in a} succinct notion. This

problem ^of being ^{able to write an} explicit ^{formula is}

bound to be faced in any attempt ^{at a} comprehensive

formalism and it will have to be faced in the present paper.

The replica method has been employed ^{in much}

more complex problems ^{than the} present, ⁱⁿ particu-

lar to spin glasses (Edwards and Anderson [5], Sherrington ^and Kirkpatrick [6]) ^{where new features}

seem to be needed and lead to complexities ^whose

status is still incomplete.

2. A field theoretic formalism.

Field theories are often presented ⁱⁿ operator form, but it is more convenient for the present purposes ^to

regard ^{them in terms} of functional integration. ^A typical ^situation arising ^{in a} network is that several chains meet and one wishes to have a formalism that

joins ^the points 01, 02, 03, 04, ^i.e.

Consider the generalization ^{of the} integral

If the indices a , f3 ^{become numerous and are} uniformly spaced, one can run this integral into the continuum to get

where

and

Provided the Dirac delta function is used to replace the Kronecker delta, the formalism transfers without

difficulty. ^{It is} often convenient to use a complex ^notation i.e. introduce a pair ^{of fields} 01,

If one considers a set of

one gets ^{zero unless n = m and then} gets

So the effect of the functional integral ^over 0, ~ ^* pairs ^{off the} points ri and ( ^{in all possible} ways. This if now g (r, r’ ) represents ^the probability ^{that a molecule} starts at r and ends at r’ then

where fl g represents all networks with an m fold functionality ^and probability g per chain between the ^cross links. Explicitly ^for example ^{if we had} just ^{two cross} links and m

3 there are only ^two types ^{of network}

available _{( i I (I I) 1}

and they ^{come from N}

3, ^M

^{2 :}

+ all permutations ^{with the} topology (i)

+ all permutations ^{with the} topology (ii)

where al, a2 ^are the numbers of ways that the ^two

diagrams ^can be constructed.

Notice that if a very large ^{number of} segments ^and

cross linkages ^are present, ^the loop diagrams ^become successively ^less important ^{and the} largest ^networks

become dominant simply by their statistical weight.

So the example ^{is not} typical ^{at all in its} weights ^but

illustrates that the integral joins up all segments ⁱⁿ all possible ways.

One may ^use the same device

to exponentiate, ^to get ^{the final} expression ^{for the}

total number of configurations

(N ^{and M are} large and therefore replace ^{N +} 1,

M + 1 for ease of writing ; ^{also N!, M! are} put ^into the normalization for the same reason).

The functional A plays the role of - H/kT ⁱⁿ

statistical mechanics. Thus the entropy ^{of a network} of M Links joining ^N chains with an m functionality

is

within the constant term from the normalisation.

Notice however that all configurations ^are permit-

ted in this integral. Thus it is not a physical network,

1677

for in a physical ^{network a} particular ^{choice of}

network is established at formation and it then becomes permanent. Nevertheless it is already ^an interesting problem ^to study ^this system ^{and will} occupy the ^next section. Since systems in statistical mechanics where all configurations ^{are accessible}

are called ergodic, this network will be so called.

It is possible ^{since A is} quadratic in 0 ^{to eliminate} cp altogether, ^{but it turns out} that this is not a fruitful

procedure, ^{and will not be further} discussed. How-

ever it is worth noting that if this is done the problem

is converted into that of

If g ^{is now} approximated by g-1=1 + p 2 k2 the

result resembles standard field theories, ^{for one has}

which frequently ^{arises in field} problems. ^Remember that > ^{and v are} fugacities ^{and so the} present problem ^is really quite different from the situation

where g, ^{v do not} appear but ^are replaced by

constants, which may involve physical properties

like the temperature, ^{but do not mark} permanent constraints as in the case here.

A version of field theory ^was used in the first formulation of polymer problems ^{in a} field theoretic formulation by Edwards and Freed [7, 8] ^{but this} original ^{work was done} before the replica ^method

was invented, and also has less flexibility ^{than the} present method, although ^{it is more} easily adapted

to random links than the present formalism. Further

developments ^{of the second} quantization ^method

have been given by ^Freed [9] ^{in a somewhat} differing

formalism from the present ^one.

3. The entropy ^{of the} ergodic ^network.

Although ^this problem ^{does not} appear physical ^at

first sight

the network obviously ^cannot just suddenly hop ^{into a new} configuration

^{all the} possible networks will be the same in the ther-

modynamic limit, ^{so the} entropy calculated will be that of a typical ^network plus ^the entropy ^{due to the} total number of ways the network ^can be made.

Given any functional integral, ^the simplest way ^to

approach ^{it is to} look for the steepest ^descent solution and expand about it. This is rather like the

approach ^{to a} quantum mechanical problem whereby

to evaluate

one takes the classical solution

and expands ^{about it,} the WBKJ method. This sometimes _{is very} good, ^{sometimes not so} good.

Fortunately ^the fully ^{cross linked} monodisperse

network turns out to be correctly treated in this

« classical » _way, so one starts by studying

Notice that the last two equations ^are always ^correct

because the system ^is thermodynamic ^{with an N-} involved ; ^{the first two are not in that} category ^{as it is}

JL, v ^not N, ^{M which} weight ^them.

The equations ^are

where ~ , ~ ^{* are the} steepest ^{descent solutions of} (3.3). ^{No bar is} required ^{on u, v as these variables}

do not fluctuate. To proceed ^to fluctuations in

0, 0 ^{* is} complicated ^{because a} problem ^{arises even}

in evaluating simple integrals ⁱⁿ Stirling’s approxi-

mation. For example, ^{if one wishes to evaluate}

the steepest ^{descent method cannot use x}

^{0 as this} is an unstable point ^{for b} > 0. Problems like this arise in an expansion ^{in cp , cp * where}

To circumvent them in the above example, ^one may

write x2

(xz) ^{+ x2 -} (X2), ^and expand ^to give

where (x2) ^{is now}

This now gives (X2) - « b -1 ₂ ^for _large ^{b and} gives ^{the correct} Stirling ^formula. This method works for cp , ^, cp ^* fluctuations. Working ^to this order

so that the numerator contains

with similar expressions ^{in 0 *.}

The equations for ~~ ^{* are now modified} by ^the

presence of cp g cp > and cp * 2 > ’ ^{and these quantities}

themselves have to be calculated self consistently. ^It

is these terms which produce fluctuations and corre-

lations in the network i.e. they quantify ^« loops » ⁱⁿ

the network. However for a well linked network these loops ^{are a} small correction proportional inversely ^{to the cross link} density ^{and we will not}

pursue them. It is interesting ^{to note} that there was

great difficulty ^{at this} point in earlier attempts [7, 8],

and therefore the authors hopes to return to a

detailed evaluation of these terms in a later paper.

From the equation ^{there is} clearly ^{a constant} solution, where, ^{since one has}

and the entropy ^is

. constants.

Thus if m

4 for example, ^{one has an} entropy ^of

which is to be interpreted ^{as a loss of one} perfect gas

degree of freedom for each cross link introduced. A system ^{cannot of course have such an} entropy ^which

means that the system ^will collapse ^{unless some}

other effect like interchain forces or entanglement repulsions ^{come into} play. ^{The model so far has not}

included these and any physical system ^{will have to.}

If one asks what will really happen ^{to a network of}

non interacting chains i.e. a phantom network, ^{it will}

take up ^{a non} uniform density, and there is indeed such a solution. Consider the equation

where g ^{is a} Gaussian, say the classic

1679

corresponding ^{to a} random walk of q steps ^of length f.

Look for a solution

Then

or in Fourier variables

i.e. ~ ^{* is a} Gaussian with coefficient times that of the chain.

This means that the network will form a ball of the order of the size of a single chain. This is clearly ^the

correct solution if it were possible ^{to have a} phantom

material. Physically, repulsive chain-chain forces and entanglements insist that a uniform density prevails, although syneresis ^can and will in general happen. The second solution will however turn out to be of great interest in the case of a set of permanent ^{cross links.}

The quadratic fluctuations correspond ^to paths ⁱⁿ

the network which close on themselves. Thus the

mean terms correspond ^{to the} assumption that any

path chosen in the network can be followed to

infinity ^{i.e. to the walls} of the material i.e. is tree like and the integral equations for ~~ ^{* can be derived} directly ^{from the tree} diagram ^without going through

the formalism.

The quadratic fluctuations allow an assessment of

loops, ^and example ^of which is shown and so on for

higher ^{terms like} cp 3, cp 4 ^{etc. An} explicit ^evaluation

of the quadratic fluctuations determinant shows that it has a small contribution for a large network and further the contribution appears ^{as a} wastage ^of links, ^a renormalisation in effect of what a segment

means. These terms will not be pursued ^{further in}

this paper.

The entropy which has been found is a function of the density of chains and cross links only. ^This

means that if other sources of free energy contribute

some Fo ^one will obtain

This system ^{has no} resistance to shear since it is

supposed that it is ergodic ^{and can} hop ^between configurations. The normal argument ^{used in net-} work theory ^{is that} Fo ^{is a} problem comparable ^to

the theory ^of liquids, ^{and is therefore not} studied for melts, although ^{it can be} usefully ^{studied in solutions}

via the Flory Huggins or successor theories. For real networks however it is argued ^that S(network) ^{now will}

contain all the description of the stress-strain re-

lationship ^{and hence, as} long ^{as one does not} study

the bulk modulus, ^will give the elastic properties ^of

the network.

To get the real network it is necessary ^to apply ^the

formalism to the case of a single permanent network,

and it will now be shown that the replica ^{method is}

available for the field theoretic formalism.

4. Replication ^{in a} field theoretic form.

The problem is that the sheared network has to have

an identical topology ^to the initial network. One can

suppose that this ^means that the network roughly keeps ^its shape ^{in the} original ^{as time} goes by, ^{and is} roughly affinely deformed when the material is deformed. Thus if the original ^{is a full line and that}

at a subsequent ^{time is} dotted, ^the expectation ^is

The simplest problem ^{is to consider} just ^this diagram : what is the joint distribution of the same

network at two remote times. The only way ^to do it is to generalize 0 (r) to cp (rl, rz), ^and replace ^the

terms of (2.21) by

and

and

If one now writes down the same A as before but in terms of the new 0, 0 ^{* with the same} u,

v structure, ^it gives ^{the identical network} repeated

in the 1 and 2 variables. The solution to the mean

equation ^{now involves} solving

Proceeding ^{as before one} may ^now obtain three different solutions. Choosing g ^{to be} Gaussian, they

are

(or ^more generally

but this has no simple interpretation). ^{The first} corresponds ^to both networks in the non uniform

condition, the third is both being uniform, ^the

second to the two networks being ^{close to one}

another in space, but the ^centre of the mass of the two being uniformly spread in the volume. If it is

argued that other bulk terms ensure that the network fills the volume and that one physically expects ^the

two networks to be close to one another it is clearly

the second solution which is the one to be con-

sidered. Moreover it is this solution which has the

maximum of entropy given ^{that the centre of mass is}

forced to be spread throughout space.

Having ^{seen what to} expect, ^the problem ^of replication ^{can be} expected ^to involve functions

and indeed if we consider

it corresponds ^to the network repeated n times in the strained volume with the identical network speci-

fication of the initial network. Thus r(O) lies in the

original volume, and the others in the strained volume.

The stationary equations ^{are now}

Again ^this equation ^{can be solved} if g is Gaussian.

The solution for the ^« ball » will be

but with uniform spread ^{of the c.m. one has}

which removes the c.m. and leaves

A convenient set of coordinates to effect the study ^of

the replica system ^is given by Deam and Ed- wards [10] ^{who use} the finite Fourier transform in the form

when our

1681

and the uniform initial distribution amounts to the absence of P ⁽⁰⁾ in 0- and 0 ^*.

The _{space of} p(O) is the diagonal ^of the cuboid of

3 (n + 1 ) ^{sides ; n} sides Al, n sides À 2’ n ^sides A 3 and 3 sides unity. This is the vector of components

The remaining p(1), Q ^{variables can be taken as} integrals ^{over an infinite} region since the whole volume is extensively larger ^than a -1/2. For example,

we can draw only ^{in two} dimensions, ^{but the} point ^is

still made clear if the x (0) is one side and x (1) the other.

The hatched area is the non vanishing region ^of

~. Our coordinates are along ^the diagonal ^{and one} perpendicular ^to it. The latter has complicated limits, ^but they ^{are remote} from the axis of large probability ^{and can} be extended to infinity.

Thus the ^« replica symmetry ^{» of} equation (1.17)

has to be broken by treating p(O) ^as radically

different from the p(1), Q in accordance to the correct physical situation. (An interesting ^discussion

of the replica symmetry ^has recently ^been given by

Goldbart and Goldenfeld [11], ^{and a} general ^refer-

ence is in the book of Mezard [12].)

All the terms in A (~, ~ * ) ^{are taken over}

) and since P(o) does not appear in 0, 4J ^{* the value} of A contains

FT ^{(1 + nÀ [)1/2 x function} independent ^{of A .}

i

Thus A

has been transformed out of the expression

3

since it is easily shown from section (3) ^{that the}

coefficient is N/2.

For non Gaussian g, the basic equation represents formidable difficulty ^if anything other than a Gaus- sian trail is take, for the n - 0 limit is not guaranteed

to be sensible, ^{and to solve} equations ^{in an} arbitrary

number of dimensions is not easy. Nevertheless

some special ^{cases will be discussed in a} later paper.

What is attractive is to try to return within the formalism to the simple ^field theory ^before taking

the limit, ^{and the next} section will suggest ^a way ^to do this.

5. A simplified ^version.

It will be seen that the experience ^{from the soluble}

Gaussian case is that it is a useful rearrangement ^to consider 0 (r(o), r(l), ^..., r(n» ^{in terms} of the collec- tive variables 0 (P(o), p(1), ^..., Q(’) > ... ) for then the average ^over all networks is equivalent ^{to the}

absence of p(O). A way ^to guarantee the absence of

p(O) if one chooses some initial trial function

~ ^{is to write}

for the integral over 03BE ^{removes the} p(O). It is therefore tempting ^to try ^a separable ^function

In general this solution will not be correct, ^{for it} restricts the form of ~ ^{which must} satisfy ^the equations (3.4) ⁱⁿ their 3 n + 3 dimensional form.

However it is possible ^to find very interesting simple

forms using ^a version of the (5.1), (5.2) ^ideas.

To do this, ^{let us introduce a variable}

Then the equations (3.4) ^are easily rearranged ^into

the form

and using ^the equations for * and * *, ^{A takes the}

value

Now try ^{for a} solution for it of

This will not in general ^be possible ^for equation (5.4)

unless the g ^{are Gaussian.} However the form does have the correct symmetry ^{and it is} interesting ^{to see}

the consequences of taking the solution to be given by (5.6) ^{where 0 is} that solution of

which has the decaying ^{structure as r - oo.}

This now allows us to produce ^an explicit form for the entropy, ^for

Hence

This formula gives ^{the Gaussian answer} immediately,

and will also give ^{an answer for the} problem ^of rods,

as is discussed in the following paper. To solve the Gaussian case try

Then

So that using

in formula ( ) ^{we obtain}

and

and, ^as usual,

Acknowledgements.

The author thanks Drs R. C. Ball, F. Boue and T. A. Vilgis ^for helpful conversations.

References

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